# Finding the angle between two straight lines | ZJ learning | Straight Lines#3

Hi everyone this is the 3rd video in our chapter

straight lines. Today we will be discussing the angle between 2 straight lines and 2 special

occasions of this angle. Lets take these 2 lines l1 and l2. Lets say

that the gradient of this line is m1 and this line m2. We can say that m1=tan(theta) and

m2=tan(beta). Let alpha be the angle between these 2 lines. So now our goal is to find

an equation for this angle ‘alpha’ Since this is a triangle we know that the

exterior angle is equal to the sum of the opposite interior angles.

So beta=alpha + theta. If we subject alpha we get alpha=beta – theta. We can also say

that tan(alpha)=tan (beta – theta). From trigonometry by using this equation tan(A

– B)=tan A – tan B/(1 + tan A*tan B), we can expand the right hand side. This gives us tan(beta) – tan(theta)/1 + tan(beta)*tan(theta).

If we substitute the gradients into this equation, we get

tan(alpha)=m2 – m1/(1 + m1m2). Notice that there are 2 angles between any

2 lines. An acute angle and an obtuse angle. We can see here that this angle ‘gamma’ is

180 – alpha. So tan(gamma)=tan(180 – alpha), so this will give us ‘-tan(alpha)’. So that

will be (m2 m1)/(1 + m1*m2). So what this means that is that if we get the value for

tan(alpha) is positive, then we have the acute angle and if the value is negative, then we

have the obtuse angle. To consider both of these angles, we put this equation in a modulus.

So this is the final equation, tan(alpha)=’modulus’ (m2 – m1)/(1 + m1*m2)

So now we can find the angle between 2 lines in terms of their gradients. Now for the tip of the day

Notice that when alpha decreases more and more and becomes zero, the 2 lines become

parallel to each other. So now we can say that tan 0=0, so (m2 – m1)/(1

+ m1*m2)=0. So then we get m1=m2. So when 2 lines are parallel the gradients are equal

to each other. Similarly when alpha equals 90 degrees, then

tan(90)=infinity. So the denominator is 0. So this gives (m2 – m1)/(1 + m1*m2)=1/0

since it is infinity. So now we can find that m1*m2=-1. So this means when the lines are

perpendicular to each other then the product of their gradients equals -1. That brings us to the end of this video

Thanks for watching and check the exercises and answers given in the description below.

Please remember to click the bell to keep updated on our new videos.

Until next time take care.